Long-Run Risk

BUSI 721, Fall 2022
JGSB, Rice University

Kerry Back

Is there risk in the long run?

A common belief is that the law of averages will rule in the long run, eliminating risk.
It’s true that the average outcome from a gamble should not be risky in the long run.

But the gain or loss from gambling is average gain per gamble \(\times\) number of gambles.

Betting on the stock market

Based on history, the bet is definitely in our favor.
Play for a long time \(\Rightarrow\) almost certainly come out ahead.
But how far ahead is quite uncertain.
- In worst 20-year period in U.S. stock market since 1926, $1 \(\rightarrow\) $1.73, a geometric average return of 2.8% per year (1929-1948).
- In best 20-year period since 1926, $1 \(\rightarrow\) $24.65, a geometric average return of 17.4% per year (1980-1999).

Simulate returns

Mean and std dev of U.S. market return 1970-2021 was 12.5% and 17.4%.
Simulate 20-year compounded returns.

import numpy as np

mn = 0.125
sd = 0.174
nyears = 20

r = np.random.normal(loc=mn, scale=sd, size=nyears)
comp_ret = np.prod(1+r)

Repeat and evaluate the distribution

nsims = 1000

r = np.random.normal(loc=mn, scale=sd, size=nyears*nsims)
r = r.reshape((nyears, nsims))
comp_ret = np.prod(1+r, axis=0)

mean     34.84
std      34.88
min       0.90
10%       7.51
25%      13.31
50%      23.92
75%      43.10
90%      75.34
max     265.18
dtype: float64

Retirement Planning Simulation

Uncertainty about long-run returns \(\Rightarrow\) uncertainty about retirement plans.

Revisit the retirement plan
Generate random returns and simulate many lifetimes.